Mean-field conditions for percolation on finite graphs

then the size of the largest component in p-bond-percolation with p = 1+O(n ) d−1 is roughly n. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n around the critical probability pc = 1 d−1 . A consequence of our theorems is that if {Gn} is a transitive expander family with girth at least ( 2 3 + ǫ) logd−1 n then {Gn} has the above scaling window around pc = 1 d−1 . In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [20] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.